Local semicircle law under weak moment conditions

Abstract

Symmetric random matrices are considered whose upper triangular entries are independent identically distributed random variables with zero mean, unit variance, and a finite moment of order 4 + δ, δ > 0. It is shown that the distances between the Stieltjes transforms of the empirical spectral distribution function and the semicircle law are of order lnn/nv, where v is the distance to the real axis in the complex plane. Applications concerning the convergence rate in probability to the semicircle law, localization of eigenvalues, and delocalization of eigenvectors are discussed.